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Pivot-minors and the Erdős-Hajnal conjecture

James Davies

Abstract

We prove a conjecture of Kim and Oum that every proper pivot-minor-closed class of graphs has the strong Erdős-Hajnal property. More precisely, for every graph $H$, there exists $ε> 0$ such that every $n$-vertex graph with no pivot-minor isomorphic to $H$ contains two sets $A, B$ of vertices such that $|A|, |B| \ge εn$ and $A$ is complete or anticomplete to $B$.

Pivot-minors and the Erdős-Hajnal conjecture

Abstract

We prove a conjecture of Kim and Oum that every proper pivot-minor-closed class of graphs has the strong Erdős-Hajnal property. More precisely, for every graph , there exists such that every -vertex graph with no pivot-minor isomorphic to contains two sets of vertices such that and is complete or anticomplete to .
Paper Structure (5 sections, 24 theorems, 18 equations)

This paper contains 5 sections, 24 theorems, 18 equations.

Table of Contents

  1. Introduction
  2. Pivot-minors and subdivisions
  3. Coherence
  4. When small balls have small mass
  5. Focused massed graphs

Key Result

Theorem 1.1

For every graph $H$, there exists $\epsilon > 0$ such that for all $n > 1$, every $n$-vertex graph not containing $H$ as a pivot-minor has two sets $A, B$ of vertices such that $|A|, |B| \geqslant \epsilon n$ and $A$ is either complete or anticomplete to $B$.

Theorems & Definitions (37)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 27 more